The role of the dissipation matrix in singular optimal control

The optimal cost in the regular stationary linear-quadratic optimal control problem is given by the maximal solution of an algebraic Riccati equation. This solution minimizes the rank of a certain matrix, which we call the dissipation matrix. The rank minimization problem is also meaningful in the singular case. Does it provide the optimal cost for the singular control problem? This question was posed by J.C. Willems in 1971. The answer is: yes.